In this paper, we are interested in the automata-theoretic approach to
synthesis, in the sense that specifications and implementations can be
defined by automata, or by automata extended with outputs called
transducers. In this approach, In and Out are
sets of words over input and output alphabets \(\Sigma \)  and \(\Gamma \) 
respectively. Perhaps the most well-known decidable instance of synthesis in
this context is the celebrated result of Büchi and
Landweber [1]}: \(\mathcal {S}\)  is the class of
\(\omega \) -regular specifications, which relates infinite input words
\(i_0i_1\dots \in \Sigma ^\omega \)  to infinite output words
\(o_0o_1\dots \in \Gamma ^\omega \)  through \(\omega \) -automata
(e.g. deterministic parity automata), in the sense that the infinite
convolution \(i_0o_0i_1o_1\dots \in (\Sigma \Gamma )^\omega \)  must be
accepted by an \(\omega \) -automaton defining the
specification. The class of implementations \(\mathcal {I}\)  is all the
functions which can be defined by Mealy machines, or equivalently,
deterministic synchronous transducers which, whenever they read some input \(i\in \Sigma \) ,
produce some output \(o\in \Gamma \)  and possibly change their own internal
state. The seminal result of Büchi and Landweber has recently triggered a
lot of research in reactive system synthesis and game theory, both on
the theoretical and practical sides, see for
instance [2]}. We identify two important limitations to the now classical setting of \(\omega \) -regular reactive
synthesis:
